Sweeping costs of planar domains

TL;DR

This paper introduces a formula for the minimal length of a continuous sensor curve needed to sweep any planar Jordan domain, linking it to geodesic Frechet distance, and identifies shapes with maximal sweeping cost.

Contribution

It provides an analytic formula for the sweeping cost of Jordan domains using geodesic Frechet distance and characterizes shapes with maximal sweeping cost.

Findings

Sweeping cost of convex domains equals their width.

Maximal sweeping cost among unit area convex domains is achieved by the equilateral triangle.

Derived an explicit formula for sweeping cost in terms of boundary curves.

Abstract

Let D be a Jordan domain in the plane. We consider a pursuit-evasion, contamination clearing, or sensor sweep problem in which the pursuer at each point in time is modeled by a continuous curve, called the sensor curve. Both time and space are continuous, and the intruders are invisible to the pursuer. Given D, what is the shortest length of a sensor curve necessary to provide a sweep of domain D, so that no continuously-moving intruder in D can avoid being hit by the curve? We define this length to be the sweeping cost of D. We provide an analytic formula for the sweeping cost of any Jordan domain in terms of the geodesic Frechet distance between two curves on the boundary of D with non-equal winding numbers. As a consequence, we show that the sweeping cost of any convex domain is equal to its width, and that a convex domain of unit area with maximal sweeping cost is the equilateral triangle.